A266402 Self-inverse permutation of natural numbers: a(n) = A064989(A030101(A003961(n))).

1, 2, 3, 4, 5, 6, 11, 8, 17, 10, 7, 12, 13, 14, 25, 38, 9, 30, 23, 20, 53, 34, 19, 36, 15, 26, 51, 28, 29, 18, 37, 76, 33, 22, 83, 24, 31, 16, 39, 40, 47, 42, 59, 46, 75, 44, 41, 218, 73, 122, 27, 52, 21, 188, 107, 56, 101, 58, 43, 100, 89, 74, 397, 152, 65, 66, 109, 134, 131, 70, 71, 514, 49, 62, 45, 32, 239, 78, 97, 120, 563, 82, 35

Offset: 1

Author

Antti Karttunen, Jan 02 2016

Keywords

• nonn
• base

Comments

Shift primes in the prime factorization of n one step towards larger primes (A003961), then reverse the binary representation of the resulting odd number (with A030101), which yields another (or same) odd number, then shift primes in the prime factorization of that second odd number one step back towards smaller primes (A064989).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000035, A003961, A030101, A057889, A064989.

Cf. A265329, A266404 (other conjugates or similar sequences derived from A057889).

Cf. also A266401, A266415, A266416.

Programs

Mathematica

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose @FactorInteger@ n; g[n_] := FromDigits[Reverse@ IntegerDigits[n, 2], 2] 2^IntegerExponent[n, 2]; h[p_?PrimeQ] := h[p] = Prime[PrimePi@ p + 1]; h[1] = 1; h[n_] := h[n] = Times @@ (h[First@ #]^Last@ # &) /@ FactorInteger@ n; Table[f@ g@ h@ n, {n, 83}] (* A266402, after Jean-François Alcover at A003961 and Ivan Neretin at A057889 *)

PARI

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A266402 = n -> A064989(A030101(A003961(n)));
for(n=1, 8191, write("b266402.txt", n, " ", A266402(n)));

Scheme

(define (A266402 n) (A064989 (A057889 (A003961 n))))

Formula

a(n) = A064989(A030101(A003961(n))) = A064989(A057889(A003961(n))).

Other identities. For all n >= 0:

A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]